Topologically informed echo state networks via poincaré return maps for chaotic time-series

Area of Science:

• Dynamical Systems and Chaos Theory
• Machine Learning and Reservoir Computing
• Computational Physics

Background:

• Accurate long-horizon forecasting of chaotic systems remains a significant challenge.
• Existing reservoir computing methods often lack formal guarantees or require extensive tuning.
• Data-driven approaches are needed to capture complex, nonlinear dynamics.

Purpose of the Study:

• To introduce the Poincaré-Section Reservoir (PSR), a novel deterministic reservoir computer.
• To demonstrate PSR's ability to forecast strongly chaotic dynamics with extended prediction horizons.
• To provide a formal consistency guarantee for data-driven chaotic system modeling.

Main Methods:

• Learning a reservoir's recurrent graph directly from system trajectory data.
• Slicing system trajectories with a hyperplane to generate symbolic sequences.
• Constructing the reservoir adjacency matrix from empirical transition frequencies.
• Proving convergence of the learned operator to the true Perron-Frobenius operator.

Main Results:

• The Poincaré-Section Reservoir (PSR) extends valid prediction time by up to 1.9x over existing methods on benchmark chaotic systems (Lorenz, Rössler, Chen-Ueta, Chua).
• PSR requires no gradient-based training, equation knowledge, or hyper-parameter tuning.
• The learned reservoir weights formally converge to the true system dynamics as the data partition is refined.

Conclusions:

• PSR offers a lean, interpretable, and fully data-driven paradigm for long-horizon forecasting of chaotic dynamics.
• The fusion of Ulam's operator discretization with reservoir computing provides a robust alternative to traditional methods.
• This method advances the state-of-the-art in predicting complex, nonlinear systems.